On time-consistent multi-horizon portfolio allocation

We analyze the problem of constructing multiple mean-variance portfolios over increasing investment horizons in stochastic interest rate markets. The traditional one-period mean-variance optimal portfolios of Hansen and Richard (1987) require the replication of two payoffs. When several maturities are considered, different payoffs have to be replicated each time, with an impact on transaction costs. Using martingale decomposition techniques and introducing a family of risk-adjusted measures linked to increasing maturities, we provide an intertemporal version of the traditional orthogonal decomposition of asset returns. This allows us to construct a multi-horizon mean-variance frontier that is time-consistent and requires the replication of solely two payoffs for all horizons under consideration. When transaction costs are taken into account, our time-consistent mean-variance frontier may outperform the traditional mean-variance optimal strategies in terms of Sharpe ratio. Some interesting examples of this fact come from long-term contracts as life annuities.